3. What is a formula?
A formula is a special type of equation that links two or more physical variables. w l
P = 2(l + w)
P represents the perimeter of a rectangle
l stands for the length of the rectangle
w symbolizes the rectangle’s width.
Teacher notes
Remind pupils that the letter symbols in formulae are called variables because their value can vary. Unlike equations where the unknown can be any real number, the variables in formulae are often limited to values determined by the context. For example, if the variable in a formula represents a length it can only take positive values. If it represents a number of objects then it can only take whole number values.
We can use this formula to work out the perimeter of any rectangle given its length and width.
We do this by substituting known values into the formula.

4. Rectangle perimeters

5. Surface area formula
The surface area, S, of a cuboid is given by the formula: w l h
S = 2lw + 2lh + 2hw
In this formula:
l represents the length of the cuboid
w is the cuboid’s width
h is the height of the cuboid.
Teacher notes
In formulae involving length, area and volume the units are not usually given. However, we must make sure that all the units are the same before substituting them into the formula.
What is the surface area of a cuboid with a length of 25 cm, a width of 32 cm and a height of 150 cm?

7. Adding values
What is the surface area of a cuboid with a length of 25 cm, a width of 32 cm and a height of 150 cm?
First, check the units are the same:
l = 25 cm, w = 32 cm and h = 150 cm h
Next, substitute the values in without the units.
S = 2lw + 2lh + 2hw
= (2 × 25 × 32) + (2 × 25 × 150) + (2 × 150 × 32)
Teacher notes
Remind pupils that 1 m2 = 1 m × 1 m = 100 cm × 100 cm = 10,000 cm2 and so establish that 18,700 cm2 is equal to 1.87 m2. The units used in the answer must be consistent with the units used in the formula. w l =
Don’t forget to add units. Here, we are calculating an area, so the units will be in the form: x².
= 18,700 cm2

8. More on formulae
Formulae deal mainly with real-life quantities such as length, mass, temperature or time and the variables often have units.
Units are usually defined when writing formulae, but they are not included in the calculations.
S = d t
For example:
This formula does not mean much unless we say:
Teacher notes
Point out that the letters representing each variable often start with the same letter as the word for the variable they are representing.
S is the average speed in m/s
d is the distance travelled in metres
t is the time taken in seconds.

9. Speed, distance and time
If a car travels 2 km in 1 min 40 secs, at what speed is the car travelling?
S = d t
Use the formula:
Write the distance and the time using the correct units before substituting them into the formula.
2 kilometres = 2000 metres 1 min and 40 secs = 100 seconds
Now substitute these numerical values into the formula.
Teacher notes
Ask pupils how we could have found the average speed of the car in km/h rather than m/s. This will be important in helping the students correctly answer the average speed questions in the next slide.
S =
2000
100
Write the units in as the final step in the calculation.
= 20 m/s

10. Speed, distance and time calculations
Teacher notes
This activity requires students to input the correct average speed. As well as testing the students’ mathematical abilities, this will also test their comprehension of a contextual problem. Each question asks for a specific type of unit in the answer, so it is important that the students read the question carefully before attempting an answer.
As an extension activity, you may wish to ask the students to work out the average speed in different units.

11. The Eiffel Tower Teacher notes
Explain that this formula is true for any object (whatever its mass) that is dropped on earth (disregarding air resistance). This is because gravity causes objects to fall with the same acceleration regardless of their mass.
Talk through each substitution.
Photo credit: © Juriee Maree, Shutterstock.com

12. Blackpool Tower
The distance, d, in metres, that an object falls after being dropped is given by the formula:
d = 4.9t2
where t is the time in seconds.
If a young boy drops a coin over the edge of the 158 m high Blackpool Tower, how far will it have dropped after 12 seconds?
Can you show how many seconds it will be before the coin hits the ground?
Teacher notes
Some students may answer that, by using the formula d = 4.9t², the coin will have travelled 4.9 × 12² = m in 12 seconds. Encourage them to question whether this is a valid answer or not. The tower is only 158 m high, so the coin cannot have fallen m. It would have hit the ground after dropping 158 m.
The second question can be viewed as an extension activity. Students should be encouraged to develop a way of finding out how long it will take before the coin hits the ground. They may try to do this through trial and error, but some students will be able to substitute d = 158 m into the equation and rearrange the formula to give them:
t = √(d ÷ 4.9)
t = √(158 ÷ 4.9)
t = seconds (6 seconds to the nearest second)
Students should be encouraged to discuss how we know for sure the answer is and not – They should realize that the time value is limited by the context.
Photo credit: © Nicholas Peter Gavin Davies, Shutterstock.com

13. Stopping distances
A formula that can be used to calculate the stopping distance of a car is:
x2 ÷ 20 + x = S
where x is equal to the car’s speed in mph and S is equal to the stopping distance in feet.
As an estimate, what will be the stopping distance of a car travelling at 30 mph?
What speed would a car have been doing if it took 315 feet to stop?
What factors might affect a car’s stopping distance?
Teacher notes
This formula can be used as an approximation of the stopping distance of a car. Use different values for x to check pupils’ substituting ability.
x² ÷ 20 + x = S
30² ÷ = S
900 ÷ = S
= S
S = 75 feet
Students should be encouraged to come up with different ways in which they could work out what speed a car had been travelling at if it took 315 feet to stop. They may choose to use trial and error to work out that, as an estimate, the car would have been travelling at 70 mph.
Various factors may affect a car’s stopping distance. These may include driver’s reaction time, weather conditions and the condition of the brakes on the vehicle. Students should realize that all of these factors will increase the stopping distance.
Photo credit: © Heather Renee, Shutterstock.com

14. Stopping distances

15. Adding numbers to formulae
Teacher notes
This activity introduces students to a variety of different formulae and gives them a brief description of what the formulae are used to calculate. Students should be encouraged to use the pen tool to show their working to calculate the correct answer. They can then press on the notelet to reveal the correct answer. Each time the reset button is pushed on this activity, a new set of conditions will be generated. This gives the activity an open-endedness that ensures it can be used several times.

16. Problems leading to equations
The formula used to find the area, A, of a trapezium with parallel sides a and b and height h is: h a b
A = (a + b)h 1 2
By substituting values into the equation and rearranging the equation, it is possible to calculate an unknown.
Teacher notes
In this example, we are told the area and the lengths of the parallel sides and we need to find the height. Explain that we can substitute the values we are given into the formula to make an equation with one unknown, h. An alternative would be to rewrite the formula with h as the subject.
What is the height of a trapezium with an area of 40 cm2 and parallel sides of length 7 cm and 9 cm?

17. Problems leading to equations
By substituting the known values into the equation, we can work out the height of the trapezium.
A = (a + b)h 1 2 h
a = 7 cm
b = 9 cm
A = 40 cm²
40 = (7 + 9)h 1 2
40 = × 16h 1 2
40 = 8h
Teacher notes
This slide shows how to solve the problem by substituting the given values to make an equation. Check the solution by substituting h = 5, a = 7 and b = 9 into the original formula to make sure that it gives an area of 40 cm2.
swap both sides
8h = 40
divide by 8
h = 5
The height of the trapezium is 5 cm.

18. Problems leading to equations
Teacher notes
This activity introduces students to a variety of different formulae and gives them a brief description of what the formulae are used to calculate. In this activity, students will need to rearrange the formulae in order to calculate the value of the unknown. Students should be encouraged to use the pen tool to show their working when calculating the correct answer. They can then press the notelet to reveal the correct answer. Each time the reset button is pushed on this activity, a new set of conditions will be generated. This gives the activity an open-endedness that ensures it can be used several times with fresh conditions.

20. Mobile phone tariffs
Around 90% of the population of the UK aged between 15 and 34 own or use a mobile phone.
Can you conduct a survey of your class to see:
how many students own a mobile
what they use it for
which brand of phone is most popular
how much people spend on their phone every month.
Teacher notes
This slide serves as an introduction to the comparisons of two different mobile phone tariffs on the next slide. This slide will give the students good practice of data collection and how best to conduct a survey. You may wish to get students to create their own questionnaires to complete this slide.
Photo credit: © robootb, Shutterstock.com
Do most people have a pay as you go phone or a contract that costs a set amount per month? Why?

21. Different costs
This graph shows the cost of two different types of mobile phone tariff with the key to the graph removed.
Which of the lines do you think represents:
a pay as you go tariff?
a monthly contract tariff?
Teacher notes
The red line represents a monthly phone contract while the green line represents a PAYG phone. On a month-to-month basis, a PAYG phone costs nothing if you don’t use it whereas a contract phone always has a monthly cost (even if the phone isn’t used at all). Obviously, there is an initial outlay of money on a PAYG phone.
Student answers will vary as to which tariff they would choose to use depending on how often they use their phone. You could ask students to combine the information on this graph with the information they collected on the previous slide to say how many people in the class should use a PAYG tariff and how many should use a monthly phone contract.
Basically, if the students are making less than 200 minutes of phone calls a month, the PAYG option is cheaper.
Can you provide a reason for your answer?
Which tariff would you choose to use? Why?

22. How much does it really cost?

23. A mobile formula Teacher notes
By reading from the graph and by using the table of values from the previous slides, students should be able to work out the formula that represents:
the monthly phone contract: C = 0.2m + 20
the PAYG phone tariff: C = 0.3m
Students should also be able to identify the point at which the phones cost the same (200 mins). As a further extension task, you could ask students a variety of questions based on this graph.
For example, if John has 6 friends that he phones every month, how long can he speak to each friend before his PAYG phone costs him more than a monthly contract?

24. Van hire Sandra is moving house and needs to hire a van for a week.
Yellow Cheapies charge £50 per day and 20 p for every mile that is driven.
Green Getters only charge £25 per day, but 40 p per mile.
Teacher notes
Yellow Cheapies: (£50 × 7) + (20 p × 100) = £350 + £20 = £370
Green Getters: (£25 × 7) + (40 p × 100) = £175 + £40 = £215.
Green Getters are £370 – £215 = £155 cheaper if Sandra hires a van for one week and travels 100 miles in the course of that week.
Encourage students to discuss how they could devise a formula for the prices charged by the two companies. Ask them to think what problems they may have to be aware of, for example, the cost will need to be calculated in pounds, but the cost per mile is currently displayed in pence.
The next slide asks the students to successfully identify which formula belongs to which company.
Which van is cheaper if she does 100 miles in the week?

25. Van hire

26. Van hire
I need a van for 5 days and I will be doing 1000 miles. Which company will be cheaper?
What’s the maximum mileage I can do in one day in a Green Getters van before I spend £75?
Teacher notes
The gentleman needs a van for 5 days during which time he will travel 1000 miles. This will cost him (£50 × 5) + (£0.20 × 1000) = £450 with Yellow Cheapies and (£25 × 5) + (£0.40 × 1000) = £525 with Green Getters. It will be cheaper for him to hire a van from Yellow Cheapies.
The lady has £75 to spend in one day with Green Getters. The hire of the van is £25, so she will have £75 – £25 = £50 to spend on mileage. This will allow her to travel £50 ÷ £0.40 = 125 miles in one day.
For the last question, if we have the formulae equalling each other, we can work out when d and m will give the same cost for both firms.
50d + 0.2m = 25d + 0.4m
50d – 25d = 0.4m – 0.2m
25d = 0.2m
125d = m
When d = 1 and m = 125, the cost is the same for both firms. The same is true when d = 2 and m = 250.
Can you find values for d and m where the cost is the same for both firms? How did you work this out?